Small scale effect on the forced vibration of a nano beam ...2Laboratory of Materials and Hydrology,...

Advances in Aircraft and Spacecraft Science, Vol. 6, No. 1 (2019) 001-018

DOI: https://doi.org/10.12989/aas.2019.6.1.001 1

Copyright © 2019 Techno-Press, Ltd. http://www.techno-press.org/?journal=aas&subpage=7 ISSN: 2287-528X (Print), 2287-5271 (Online)

Small-scale effect on the forced vibration of a nano beam embedded an elastic medium using nonlocal elasticity theory

Samir Belmahi1, 2, Mohammed Zidour1, 3 and Mustapha Meradjah2

1Department of Civil Engineering ,University of Ibn Khaldoun, PB 78 Zaaroura, 14000 Tiaret, Algeria.

2Laboratory of Materials and Hydrology, University of Djillali Liabes,Sidi Bel Abbés, Algeria 3Laboratory of Geomatics and Sustainable Development, University of Ibn Khaldoun Tiaret, Algeria

(Received May 7, 2018, Revised August 4, 2018, Accepted August 6, 2018)

Abstract. This present article represents the study of the forced vibration of nanobeam of a single-walled carbon nanotube (SWCNTs) surrounded by a polymer matrix. The modeling was done according to the Euler-Bernoulli beam model and with the application of the non-local continuum or elasticity theory. Particulars cases of the local elasticity theory have also been studied for comparison. This model takes into account the different effects of the interaction of the Winkler’s type elastic medium with the nanobeam of carbon nanotubes. Then, a study of the influence of the amplitude distribution and the frequency was made by variation of some parameters such as (scale effect (e0

a), the dimensional ratio or aspect ratio (L/d), also, bound to the mode number (N) and the effect of the stiffness of elastic medium (Kw). The results obtained indicate the dependence of the variation of the amplitude and the frequency with the different parameters of the model, besides they prove the local effect of the stresses.

Keywords: carbon nanotube; nanocomposite; nanobeam; vibration; Euler-Bernoulli; Winkler

1. Introduction

The use of carbon nanotubes in the manufacturing of nanocomposites is one of the most

important topics now in materials sciences. Since their discoveries by Sumio Iijima in 1991 (Iijima

1991) are a fundamental part of nanotechnology studies (Ahmadi Asoor et al. 2016). Carbon

nanotubes have attracted many research activities, which are closely related to their extraordinary

electrical, mechanical, thermal, physical and chemical properties (Ahmadi Asoor et al. 2016, Kiani

2014a). Carbon nanotube technologies have many applications for to the aerospace industry. It is

an advanced composite material of the future in aerospace industry (Mrazova, 2013, Harris 2009).

Their use is based on the knowledge at different scales, of their behavior and characteristics

(Hurang et al. 2010) considering the different factors which are involved (Shehata et al. 2011).

The nanocomposites are materials with a nanometric structure (Acton 2013, Okpala 2014) on a

scale that lies between 1 and 100 nm (de Azeredo et al. 2009). They have the capacity to improve

the macroscopic properties of the products (Okpala 2013) as well as the mechanical properties

(Sachse et al. 2013), without compromising the ductility of the material (Bakis et al. 2002). In

fact, the small size of these particles does not create a large concentration of constraints (Okpala

Corresponding author, Ph.D., E-mail: [email protected]

Samir Belmahi, Mohammed Zidour and Mustapha Meradjah

2014).

Moreover, they are considered as heterogeneous microscopically and homogenous

macroscopically (Gay 1987). The cohesion of the nanocomposite is ensured by the matrix or the

charge which can be in the general case a polymer or an organic material (Hurang et al. 2010),

while the carbon nanotube represents the reinforcement (Okpala 2013, Öchsner et al. 2013) which

is responsible for these mechanical performances and the overall stability (Hurang et al. 2010).

The carbon nanotube increases the stiffness which implies the increase of the Young’s modulus

(Kaushik et al. 2015) and the breaking constraint of the nanocomposite (Choi et al. 2005). It

represents an allotropic form of carbon distinct from diamond and graphite (Kaushik et al. 2015)

and has only one layer of graphene wound on itself (Harris 1999). CNTs are ultimate reinforcing

agent, called nanofibers, in different matrix materials for the development of a new class of

nanocomposites that are extremely strong and ultra-light (Kumar et al. 2016). So the polymers

consist of hydrocarbon macromolecular chains (Okpala 2014, Kumar et al. 2003) of different

lengths and very high stability (Hina et al. 2014).

Carbon nanotubes are currently of interest in several fields of mechanical engineering and for

construction. They are currently the basis of most composites materials, so their behavior in a

matrix will be linked to this last and the results will be completely different. Several questions

have been elaborated on this material, in particularly the free vibration, among which we quote

some works on the linear and the non-linear vibration analysis of longitudinal vibration: By use

and application of different theories: Ahmadi Asoor et al. (2016), Investigation on vibration of

single-walled carbon nanotubes by variational iteration method. Togun et al. (2016), Nonlinear

free vibration of a nanobeam based on non-local Euler-Bernoulli Beam theory, Ansari et al.

(2013), Nonlinear finite element vibration analysis of DWCNT based on Timoshenko beam theory.

Flexural free vibration and buckling analysis of SWCNT using different gradient elasticity theories

(Karlicić et al. 2015, Rakrak et al. 2016). by embedded or related the NTC at different medium:

Sound wave and free vibration of CNT embedded in different elastic medium using nonlocal

elasticity theory and finite element formulation (Gafour et al. 2013, Togun et al. 2016, Nguyen et

al. 2017 Karami et al. 2018a, Dihaj et al. 2018, Chemi et al. 2018, Hamidi et al. 2018), Analysis

of nonlinear free vibration for DWCNT by Hajnayeb et al. (2015), Nonlinear free vibration of

SWCNT embedded in viscoelastic medium by Ali-Akbari et al. (2015) and nonlinear vibration

analysis of the Fluid-Filled SWCNT with the Shell model based on the nonlocal elasticity theory

by Soltani et al. (2015).

Recently, the continuum mechanics approach has been widely and successfully used to study

the responses of nanostructures, such as the static (Yazid et al. 2018, Ahouel et al. 2016, Zemri et

al. 2015, Karami et al. 2017, Karami et al. 2018b), the buckling (Khetir et al. 2017, Bellifa et al.

2017a, Khetir et al. 2017, Houari et al. 2018), free vibration (Bounouara et al. 2016, Mouffoki et

al. 2017, Besseghier et al. 2017, Al-Basyouni et al. 2015), Dynamic analysis, wave propagation

and free vibration (Karami et al. 2018c, Behrouz et al. 2016, Ait Yahia et al. 2015, Behrouz et al.

2018, Besseghier et al. 2017, Bouafia et al. 2017, Belkorissat et al. 2015, Youcef et al. 2018),

shear deformation theory by Mokhtar et al. (2018).

Also, these studies have been started at different scales on composite structures including FGM

plates: By using different theories of deformation: The new theories of vibration, bending and

wave propagation, applied on shear deformation of FGM and laminated composite plates (Mahi et

al. 2015, Bellifa et al. 2016, Ait Yahia et al. 2015, Boukhari et al. 2016) and for refined plate with

four variable (Bellifa et al. 2017b, Fourn et al. 2018) and shells by Zine et al. (2018). by new shear

deformation theories using only two and three variables: 3-unknown hyperbolic shear deformation

2

Small-scale effect on the forced vibration of a nano beam embedded an elastic medium…

theory for vibration of functionally graded sandwich plate by Belabed et al. (2018); shear

deformation theory for buckling analysis of single layer graphene sheet based on nonlocal

elasticity theory by Mokhtar et al. (2018); Hachemi et al. (2017) three-unknown shear deformation

theory for bending analysis of FG plates resting on elastic foundations and Mouffoki et al. (2017)

nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear

deformation beam theory.

Studies on thermal stability of plates and functionally graded sandwich plates (Bousahla et al.

2016, Bouderba et al. 2016, Menasria et al. 2017). Thermal buckling (El-Haina et al. 2017, Chikh

et al. 2017); thermoelastic analysis (Tounsi et al. 2013, Attia et al. 2018), thermomechanical

bending response by Bouderba et al. (2013). Hygro-thermo-mécanique (Zidi et al. 2014, Beldjelili

et al. 2016). Thickness stretching effect for the flexure analysis of laminated composite plates

(Draiche et al. 2016, Bouhadra et al. 2018); stretching effect for thermomechanical bending by

Hamidi et al. (2015). Quasi-3D and 2D shear deformation theories (Younsi et al. 2018 and Hebali

et al. 2014). Bending and free vibration analysis of FGM and composite plates by Abualnour et al.

(2018); sinusoidal shear deformation theory for functionally graded plates by Benchohra et al.

(2018). Nonlocal strain gradient 3D elasticity theory and nonlocal strain gradient higher order shell

theory by Karami et al. (2018a, b). new theories for vibration analysis of five variable of refined

plate by Bennoun et al. (2016) and higher order shear for composite plates by Bousahla et al.

(2014) and Belabed et al. (2014).

Recent studies and applications in mechanics have shown the need to study boundary

conditions. They are usually used to evaluate constants of integration when you are performing an

indefinite integral. Abdelaziz et al. (2017) utilized various boundary conditions for bending,

buckling and free vibration of FGM sandwich plates. A simple refined theory for buckling and free

vibration of exponentially graded sandwich plates under various boundary conditions is studied by

Ait Amar Meziane et al. (2014). Post-buckling analysis of shear-deformable composite beams

using a novel simple two-unknown beam theory has been studied by Kaci et al. (2018).

It is clear from this summary above, that works on nanoscale materials are being researched

compared to materials at other scales of structure and nanostructure. In particular we have seen

that most studies are done under a free vibration hence the importance of studying in this

document the forced vibration. This carbon nanotube (CNT) will be integrated in a polymer matrix

that represents the elastic medium according to the Winkler model. The nanobeam is represented

by the Euler-Bernoulli beam and by application of the non-local elasticity theory. The carbon

nanotube considered in this work is of the single-walled (SWCNTs).

2. Modeling of the elastic medium

The combination of the nanocomposite indicated above, allowed us to use mechanical

modeling where it is supposed that the nanotube is a material embedded on an elastic medium

equivalent to the polymer matrix. For this study, several approaches exist, such as Winkler’s one in

1867, but also Filonenko and Borodich in 1940, Hetényi in 1950, Winkler-Pasternak in 1954,

Vlasov in 1960 and Kerr in 1964 (Iancu 2009a, Mourelatos 1987, Rajpurohit et al. 2014,

Selvadurai et al. 1979). Historically we can say that the Winkler model used without this study has

proved its efficiency: Starting with Boussinesqin 1885 who studied the problem of a semi-infinite,

homogeneous, isotropic, linear and elastic medium submitted to a concentrated vertical load (P)

(Selvadurai et al. 1979). Vesic in 1961 compared the results of an infinite beam on an elastic

3


Fig. 1 Deflections of elastic foundations under an uniform load (Mourelatos 1987, Iancu et al. 2008): a.

Winkler foundation; b. real soil foundations

Fig. 2 Winkler model (a) any load; (b) concentrated load, (c) rigid foundation, (d) flexible foundation.

(According to Selvadurai, 1979) (Selvadurai et al. 1979)

foundation (Chandra et al. 1987). Gibson in 1967, confirmed the displacement discontinuity which

appears between the loaded portion and the unloaded portion of the foundation surface (Gibson-

soil) (Figure 1 and 2 (a and b)), where he determined that displacements are almost constant below

the loaded area and are negligible outside this zone and that for different cases of loadings (see

Figs. 1(a)-(b)) (Iancu 2009a, Chandra et al. 1987).

Winkler supposes in his modeling a beam placed on an elastic support or foundation, modeled

by a series of vertical springs, identical, infinitely close, without coupling effect, linearly elastic

and with a stiffness noted kw(x) (Iancu 2009a, Hetenyi 1961, Karasin et al. 2014). This model does

not consider the transverse shear deformations to simplify the resolution and the obtaining of the

analytical solutions (Hetenyi 1961). The discontinuity of the Gibson is validated for this model

(see Fig.1) and on the other hand, the settlements of the loaded area in the case of a rigid

foundation remain the same in the case of a flexible foundation (see Figs. 2 (c)-(d)) (Selvadurai

1979). This approach presents a linear relationship between the normal algebraic displacement of

the structure and the contact pressure between the beam and the elastic foundation (Gorbunov-

Posadov et al. 1973). Subsequently, Pasternak in 1954 introduced the shearing interaction between

the springs by connecting the ends of these last ones to a layer made of incompressible vertical

elements that deform only by transverse shear (Mourelatos 1987, Selvadurai et al. 1979, Dinev

2012). This is the Winkler-Pasternak model.

4


Fig. 3 Geometries and arrangement of a carbon nanotubes incorporated in an elastic medium

The stiffness kw(x) indicated above is named by the proportionality constant (Iancu 2009b) and

known as the support soil reaction module (Selvadurai et al. 1979, Iancu et al. 2008). In the case

of a linear carbon nanotube (Karasin et al. 2014) with a constant stiffness (Kacar et al. 2011), we

can write:

)()( xwkxf w−= (1)

Where w(x) is the transverse displacement of the nanobeam in the z direction and kw is the

Winkler foundation modulus.

3. Problem formulation

Considering homogeneous nanobeam of single-walled carbon nanotube with a length (L),

thickness (t) a constant section (A) and a density (ρ). The nanobeam is incorporated an elastic

medium with stiffness kwin (fig. 3).

Hooke’s law for a uni-axial state of stress is given by the following equation (Heireche et al.

2008):

( ) )()(

)(2

22

0 xEx

xaex

=

−

(2)

Where E is the modulus of elasticity for the constitutive material of the element; e0a, is the

scale effect (a: is an internal characteristic length which represents the length of a C-C bond, and

e0 is an adjustable parameter). ε(x) is the strain given as:

x

xwyx

2

2 )()(

−=

(3)

The equation of motion of a monolayer carbon nanotube given by Doyle (1997) from which the

loading and the elastic medium are added:

5


0)(

)()()(

2

2

=

−++

t

xwAxfxq

x

xV

(4)

Where w(x) is the displacement in the z-direction, (ρA) is the mass per length of nanobeam,

V(x) is the shear force, q(x) is the distributed load on the nanobeam and f(x) is the interaction

pressure per unite axial length between the nanotube and the surrounding elastic medium

represented by equation (1).

From the classical theories of material resistance, the resultant of the shear force V(x) on the

cross-section of the nanotube is equal to the derivative of the bending moment with respect to the

variable x:

x

xMxV

=

)()(

(5)

And the resulting bending moment M(x) in a nanobeam section is given as follows:

=A

dAyxM )(

(6)

From Eqs. (2)-(3)-(4)-(5) and (6) the bending moment is equal to:

( ) 0)]()()(

[)(

)(2

22

02

2

=−−

+

−= xfxq

t

xwAae

x

xwEIxM

(7)

Where I is the moment of inertia for the cross section of the nanobeam.

From Eqs. (5) and (7) the shear force is equal to:

( )( )

])()(

[)(

)(2

32

03

3

x

xf

x

xq

tx

xwAae

x

xwEIxV

−

−

+

−=

(8)

By substituting Eq. (8) into Eq. (4), we obtain the unidimensional general differential equation

of the forced vibration of a monolayer carbon nanotube based on the Euler-Bernoulli theory and

following the Winkler approach:

( ) 0))()()(

)((1)(

2

2

2

22

04

4

=−−

−+

xfxq

t

xwA

xae

x

xwEI

(9)

Substituting the response of the support f(x) equivalent to Eq. (1) we obtain the governing

differential equation (Eq. 10) of forced vibration for the Euler-Bernoulli beam in an elastic

medium:

( ) 0))()()(

)((1)(

2

2

2

22

04

4

=+−

−+

xwkxq

t

xwA

xae

x

xwEI w

(10)

In the case of the local model (e0a = 0), not forced (free vibration: (q(x) = 0) ), Eq. (10) will

represent the governing differential equation of free vibration for the Euler-Bernoulli beam (Kacar

et al. 2011) and is expressed as follows:

6


0)()()(

2

2

4

4

=+

+

xwk

t

xwA

x

xwEI w

lx 0 (11)

Returning to Eq. (10), the resolution of the latter with partial derivatives of order four is done

by the technique of separation of the variables (x and t). Knowing we treat a vibration problem, it

will be assumed that displacement w(x, t) and loading q(x, t) are sinusoidal functions of pulsating

(ω) (Heireche et al. 2008).

So we pose:

=

=txqtxq

txWtxw

m

NL

sinsin),(

sinsin),(

,....)2,1( == ml

N

(12)

The resolution of this differential Eq. (10) gives the amplitude in the model of non-local

elasticity theory, WNL(x) such as:

woow

mNL

kLaeLAaeLkLALEI

aeLqxW

422422244244

20

24

)()(

))(1()(

+−+−

+=

(13)

)()(

24

4

−+=

EI

LqxW m

NL

(13a)

With

=

=

=

+=

EI

Lk

EI

LA

L

ae

w4

422

20

2 )(1

(14)

In the case of local model (e0a = 0), the amplitude WL(x) will be expressed as follows:

)()(

24

4

−+=

EI

LqxW m

L

(15)

For a free vibration where (q(x) = 0), without elastic medium (kwin = 0) and in the case of the

model of local elasticity theory (e0a = 0), from Eq. (13) we have:

024 =− nAEI

(16)

Where:

A

EIn

42 =

(17)

7


Table 1 Results of frequency and amplitude ratios as a function of dimensional ratios (L / d) with an elastic

medium and a constant mode number (N = 6)

r L/d=10 L/d=20 L/d=30

2 0.930189 0.977463 0.988388

2.5 0.958901 0.987121 0.993678

3 0.972650 0.991548 0.995939

3.5 0.980399 0.993990 0.997145

4 0.985228 0.995492 0.997874

4.5 0.988452 0.996487 0.998351

5 0.990717 0.997182 0.998682

5.5 0.992370 0.997688 0.998921

6 0.993616 0.998068 0.999099

6.5 0.994578 0.998360 0.999237

7 0.995337 0.998591 0.999345

8 0.996443 0.998926 0.999501

9 0.997197 0.999154 0.999608

10 0.997734 0.999317 0.999683

In the Eq. (14) we have:

EI

LA

422

=

(18)

By substituting the Eq. (17) into Eq. (18) we have:

242 r= (18a)

With:

2

22

n

r

=

(19)

So, (r) is the frequency ratio between the forced vibration and the free vibration. To study the

effect of locality of the constraints or loading, the ratio between the two amplitudes non-local

WNL(x) and local WL(x) is calculated and worths:

+−

+−=

)1(

)1(24

24

r

r

W

W

L

NL

(20)

4. Results and discussion

The characteristics of the carbon nanotube used in the calculation are as follows: the modulus

of young E = 1TPa, the Poisson’s ratio υ = 0.19, the mass density ρ = 2.3 g /cm3 and the effective

thickness (t) of NTC is taken equal to 0.3 nm.

8


To better interpret the results obtained in the calculation during this work, the calculated values

are grouped in tables and represented by the figures. So, the Tables 1-3 represent the values of the

ratios of the amplitudes between the non-local and local elasticity theory (WNL / WL) computed as a

function of the ratios (r) between the forced and free frequencies. In Table 1, the effect of the

dimensional ratio (L/d) is shown in Fig. 5.

In Table 2, we presented the effect of the mode number (N) shown in Fig. 6.

Table 2 Results of the ratios of frequencies and amplitudes as a function of the mode number with an elastic

medium and constant dimensional ratio (L / d = 5)

r N=1 N=3 N=6

2 0.988388 0.930189 0.848146

2.5 0.993678 0.958901 0.907189

3 0.995939 0.972650 0.937087

3.5 0.997145 0.980399 0.954434

4 0.997874 0.985228 0.965432

4.5 0.998351 0.988452 0.972856

5 0.998682 0.990717 0.978111

5.5 0.998921 0.992370 0.981969

6 0.999099 0.993616 0.984886

6.5 0.999237 0.994578 0.987147

7 0.999345 0.995337 0.988934

8 0.999501 0.996443 0.991547

9 0.999608 0.997197 0.993331

10 0.999683 0.997734 0.994604

Table 3 Results of the ratios of frequencies and amplitudes as a function of the parameter of the stiffness of

elastic medium and with a mode number and dimensional ratio constants (N= 6; L/d= 10)

r without elastic medium with elastic medium

2 0.994099 0.990800

2.5 0.996620 0.995745

3 0.997779 0.997432

3.5 0.998419 0.998251

4 0.998814 0.998722

4.5 0.999075 0.999020

5 0.999258 0.999223

5.5 0.999391 0.999368

6 0.999491 0.999475

6.5 0.999568 0.999556

7 0.999629 0.999620

8 0.999717 0.999712

9 0.999777 0.999774

10 0.999820 0.999818

9


and in the Table 3, the effect of the elastic medium (kwin), is shown in Fig. 7. A case is shown in

Fig. 4, for the constant values (L/d=10, N=6 and without medium elastic).

The Tables 4-5 group successively, summarizes the calculated values of the effect of the

dimensional ratio (L/d) and the effect of the mode number (N) on the ratio of the local and non-

local amplitudes (WNL/WL) for both cases of elastic and inelastic medium. These results are shown

successively in Fig. 8 and Fig. 9.

Table 4 Results of the ratios of amplitudes, according to parameters of dimensional ratios in different cases

of stiffness of elastic medium)

Dimensionnel Ratio (L/d) without elastic medium with elastic medium

10 0.985232 0.985228

20 0.995512 0 .995492

30 0.997922 0 .997874

40 0.998814 0 .998722

50 0.999235 0 .999072

60 0.999467 0 .999161

Table 5 Results of the ratios of amplitudes, according to parameters of mode number in different cases of

stiffness of elastic medium)

Numer of mode (N) without elastic medium with elastic medium

1 0.999001 0.996838

2 0.996112 0 .995939

3 0.991619 0 .991548

4 0.985927 0 .985890

5 0.979473 0 .979451

6 0.972664 0 .972649

Fig. 4 The relationship between the amplitudes ratio (WNL/WL) and the frequency ratio (r)

2 3 4 5 6 7 8 9 10

0,975

0,980

0,985

0,990

0,995

1,000

WN

L/W

L

r

10


Fig. 5 The relationship between the amplitudes ratio (WNL/WL) and the frequency ratio (r) for different

dimensional ratios (L/d) and mode number constant (N=6)

Fig. 6 The relationship between the amplitudes ratio (WNL/WL) and the frequency ratio (r) for different

mode number (N) and dimensional ratio constant (L/d= 10)

4.1 Effects of the parameters related to the CNT

The dependence of the ratio of the amplitude (WNL/WL) with the frequency ratio (r) of the

carbon nanotubes located in an elastic medium is illustrate in the Figs. 4-6.

The ratio (WNL/WL) will be an index to quantitatively evaluate the scale effect on solutions of

carbon nanotube vibrations. Firstly, it is found in the Fig.4 that this variation is important for any

frequency value r ≤ 5, from this value; the dependence becomes very low and begins to stabilize

with amplitude close to unity. It means that for the highest forced vibration frequencies (r > 5, in

our case) we can neglect scale effect and we only based on the local elasticity theory. Thereafter,

we see in Fig. 5, that the amplitude (WNL/WL) has its maximum for the weakest dimensional ratio

2 3 4 5 6 7 8 9 10

0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1,00

L/d=10

L/d=20

L/d=30

WN

L/W

L

r

2 3 4 5 6 7 8 9 10

0,84

0,86

0,88

0,90

0,92

0,94

0,96

0,98

1,00

N=1

N=3

N=6

WN

L/W

L

r

11


Fig. 7 The relationship between the amplitudes ratio (WNL/WL) and the frequency ratio (r) with and

without elastic mediums for the values (N=6 and L/d= 10)

Fig. 8 The relationship between the amplitudes ratio (WNL/WL) and the dimensional ratio (L/d) with and

without elastic mediums, (N=6)

Fig. 9 The relationship between the amplitudes ratio (WNL/WL) and number of mode (N) with and without

elastic mediums, (L/d=10)

2 3 4 5 6 7 8 9 10

0,9900

0,9915

0,9930

0,9945

0,9960

0,9975

0,9990

1,0005

Without elastic medium

with elastic medium

WN

L/W

L

r

10 15 20 25 30 35 40 45 50 55 60

0,984

0,986

0,988

0,990

0,992

0,994

0,996

0,998

1,000

1,002

Without elastic medium

with elastic medium

WN

L/W

L

L/d

1 2 3 4 5 6

0,970

0,975

0,980

0,985

0,990

0,995

1,000 without elastic medium

with elastic medium

WN

L / W

L

N

12


(L/d), equivalent at the shortest nanotube knowing that the diameter (d) is taken constant, as well

as in the Fig.6, the amplitude has its maximum for the biggest mode, hence the importance of the

use of non-local elasticity theory and vice versa.

4.2 Effect of the elastic medium

The Figs. 7-9, illustrate the effect of the elastic medium on the amplitude of the carbon

nanotube with the different scale or dimensional ratio (L/d) and the modes number (N).

It can be said that the amplitude of the nanotubes which represents the relationship between the

effect of the local and non local continuum theory, depends on the nature of the medium where it is

located (see Fig. 7), this dependency is low in the case of the non elastic medium compared to

another elastic medium and this for frequency ratios r ≤ 5; this means that the use of non-local

elasticity theory is necessary for the high stiffness. For frequency ratios r > 5 the effect of the

nature of the elastic medium becomes negligible. Also, it is still evident that the use of non-local

elasticity theory is necessary for the case of the weakest dimensional ratios (L/d ≤ 50, in this

study) and that for any values of the stiffness (kwin) of the medium (see Fig. 8). From the value L/d

> 50, the local effect is important with a small variation between an elastic medium and a non-

elastic medium. The use of non-local elasticity theory is also necessary for the higher modes (N),

from the second mode in our study (see Fig. 9) and this for any values of the stiffness (kwin) of

medium.

5. Conclusion

In this paper, the study of a single-wall carbon nanotube element (SWCNTs) represented by a

Euler-Bernoulli beam model, subjected to a forced vibration and surrounded by a winkler-type

elastic medium was presented. Numerous parameters were taken into account including, the

dimensional ratio (L/d), the mode number (N) and the stiffness of the elastic medium (kwin). The

results have shown the dependence of the vibration amplitudes on the different parameters cited

above. Moreover, it is found that the application of the nonlocal elasticity theory model or theory

is important for high mode numbers and short nanotubes. Also, it can be concluded in the case of

high rigidities than the non-local theory is necessary.

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